Something is wrong with "Link object data" in Blender 2.81
Set Origins First.

Speculate that the array objects are the result of applying array modifier and separate by loose parts. 

Notice they all share the origin of bottom left corner original.

When you link data the non offset mesh of active is given to all objects of array, hence you have all objects in same location.

Simply set origin to geometry of array selection (eg in question GIF, after selecting all white spheres, and  before selecting red object and linking object data.